W dimer diffusion on W(110) and (211) surfaces

W dimer diffusion on W(110) and (211) surfaces

ii!ii~ i~iii!~i!iii i surface science ELSEVIER Surface Science 339 (1995) 247-257 W dimer diffusion on W(ll0) and (211)surfaces Wei Xu *, James B...

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ii!ii~

i~iii!~i!iii i

surface science ELSEVIER

Surface Science 339 (1995) 247-257

W dimer diffusion on W(ll0) and (211)surfaces Wei Xu *, James B. Adams Department of Materials Science and Engineering, University of lllinois at Urbana-Champaign, 105 South Goodwin Avenue, Urbana, IL 61801, USA

Received 10 January 1995; accepted for publication 17 May 1995

Abstract Tungsten (W) dimer self-diffusion on the W(ll0) and (211) surfaces is investigated using a fourth moment approximation to tight-binding theory. For the (110) surface, we find the optimal diffusion mechanism is a concerted jump, i.e., adatom pairs hopping simultaneously on the surface. The linear displacement of a W dimer in the (111) direction is energetically favored over orientation changes. On the (211) surface, there are two different diffusion mechanisms. For a dimer in one channel along the (111) direction, the mechanism is a concerted jump with a high activation energy. For adatoms located in adjacent channels, dimers diffuse mainly via individual adatom jumps, and such dimer diffusion is highly correlated. For the present calculations, the activation energies for W dimer migration are similar to those for monomer diffusion, in good agreement with field ion microscopy (FIM) observations. The diffusion mechanism and diffusion anisotropy can be understood with a simple bond coordination model. Keywords: Clusters; Computer simulations; Construction and use of effective interatomic interacions; Diffusion and migration; Single

crystal surfaces; Surface diffusion; Tungsten

1. Introduction The migration of small clusters adsorbed on metal surfaces is important to understand crystal growth processes, the formation of thin films, and other surface-related phenomena. Experimentally, the self-diffusion of W dimers on W surfaces has been extensively investigated by field ion microscopy (FIM) [1-7]. For W dimer diffusion on the W ( l l 0 ) surface, FIM studies found that adatom pairs are closely packed along the (111) directions, and generally maintain a constant bond length while they

* Corresponding author. Fax: +1 217 244 6917; E-mail: w_ [email protected].

diffuse in the (111) direction. The activation energy is similar to that for monomers. At slightly higher temperatures, the dimer can change its orientation. However, linear displacements of W dimers on the (110) are more common than orientation changes [7]. On the W(211) surface, the two adatoms of a dimer may occupy sites in adjacent channels ("cross-channel" pairs), and diffuse parallel to the channels [2,5,7]. Two bond lengths have been identified: (1) 4.47/~ with the bond orientation perpendicular to the channel rows or (2) 5.24 ,~ with the angle between bond direction and the channel rows being 58.5 ° [3,7]. Also FIM studies found that the activation energy of the W dimer is similar to that for single adatom diffusion. Diffusion of the W dimer on the (211) will be either a "dumb-bell" flipping

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248

W. Xu, J.B. Adams~Surface Science 339 (1995) 247-257

process or a simultaneous jumping mechanism accomplished by hops of individual adatoms. In contrast to adatom pairs in adjacent channels, dimers formed in the same channel of the (211) surface are much less mobile than monomers. The closely bound W dimers in one channel make occasional jumps only above 420 K [7] and their migration has not been studied in detail. Despite great experimental interest, theoretical studies for W dimer diffusion on W surfaces are rare, mainly due to the lack of reliable interatomic potentials and large computational resources, Small cluster migration of fcc metal surfaces has been studied recently using embedded atom method (EAM) type potentials [8-11] and other theoretical models [12]. However, the intrinsic differences of fcc and bcc structures can result in very different trends in surface diffusion. For example, single adatoms migrate most rapidly on fcc crystal dense-packed (111) surface, but this diffusion trend is reversed for bcc metals, where the activation energy of single adatom diffusion on the dense packed surface (110) is slightly higher than on other open surfaces like (211) and (321) [13,16]. In this paper, we studied the self-diffusion of W dimers on the (110) and (211) surfaces, using a fourth moment approximation to tight-binding theory. In previous papers [14-16], this fourth moment method has successfully yielded the W(100) (v~× v~-)R45 ° surface reconstruction, and reasonable activation energies for single adatom self-diffusion on the W(ll0), (211) and (321) surfaces. This paper is arranged as follows. The theoretical approach used in this work will be described in Section 2. The results of W dimer self-diffusion on the (110) and (211) surfaces will be reported in Section 3. Comparisons are made to experimental results, followed by discussion. Finally, conclusions will be drawn in Section 4.

2. Theoretical model

2.1. The fourth moment method In this work we used a theoretical model which is based on the low-order moment approximation to

tight-binding theory (the fourth moment method). The total energy is given by 1 E=-~EVpair(Rij)+E(2'+E(3)+E (4).

(1)

i,j

Here Vpair(Rij) is a pair interaction between pairs of ions separated by a displacement R O. The E (z), E (3) and E (4) are the second, third and fourth moment term of the electronic density of states which are evaluated by taking double, triple and quadruple products of tight-binding matrix elements of the d-orbitals. These moment terms represent the angular components of the interatomic interactions. The details of this potential have been described elsewhere [14], therefore we will not repeat it here. The parameters in this fourth moment method were determined empirically by fitting to several measured bulk properties, namely the equilibrium lattice constant, the cohesive energy, the elastic constants, vacancy properties and four zone edge phonons. This model was previously used to study surface structure and surface diffusion [15,16,18]. Generally, the calculated results are in good agreement with experimental studies and first principle calculations. The 4th moment term (an angular term) was found to be particularly important in predicting surface structures, like the (100) reconstruction. Without the 4th moment term, this model is essentially an embedded-atomtype model, since the 3rd moment term is small.

2.2. Computational method The W ( l l 0 ) and (211) surfaces are modelled by creating slabs with two free surfaces and periodic boundary conditions in the two directions parallel to the surface. The periodic length is held fixed at the bulk value. The N-atom slabs are 20-30 layers thick with 416 and 552 total atoms for the (110) and (211) systems, respectively. Energy minimization is used to determined the stable structures of the (110) and (211) surfaces (the total energy of the system is denoted as Esub). In general, the diffusion coefficient D can be written in an Arrhenius form:

(Ed)

D = D 0exp - - ~

,

(2)

W. Xu, J.B. Adams~Surface Science 339 (1995) 247-257

where T is the temperature in Kelvin, K is the Boltzmann constant, E d is the activation energy and D O is the pre-exponential factor. Within the framework of transition state theory [19,20], we can determine the activation energy E d for a jump between two nearest stable sites. In this case, two adatoms are put on top of the fully relaxed surfaces and relaxed freely to determine the ground state of each configuration (Emin), allowing all atoms to relax. Then one adatom is marched towards another minimum potential site, allowing it to relax in the plane perpendicular to the vector between the two stable sites. This will ensure finding the minimum (optimal) energy path for adatom migration. Meanwhile, all other atoms (including the other adatom of a dimer) are fully relaxed, except atoms on the bottom layer which are frozen to prevent the shift of the simulation cell. One stationary point (maximum) is found and it corresponds to the adatom at the saddle point (Esad). The activation energy E d is given by E d = Esa d - Emi n •

(3)

3. Results and discussion

3.1. W dimer diffusion on W(llO) Fig. 1 shows four possible paths for W dimer diffusion on the (110) surface. The orientation of the dimers in Fig. 1 is along the [111] direction. Path 1 displays the adatom pair hops along the [111] direction with the bond orientation parallel to the diffu-

0

249

0

0

0

0

0

0

1

O

0

0

0 0

0

0

0

0

0 [001]

0 0

[110]

0

0 0

0 0

O

0

~- ~101

Fig. 1. Schematic of W dimer diffusions on W ( l l 0 ) surface. Adatom pairs are closely bound along the (111) direction with four possible migration paths. Here open and dark circles indicate the top layer atoms and adatoms, respectively.

sion path. For paths 2 and 3, the dimers rotate their bond directions to the [111]. The difference between path 2 and path 3 is that the bond directions at the bridge sites are [110] and [001], respectively. For path 4 the adatom pair diffuse along the [111] simultaneously. From the present calculations, two obvious features can be observed for W dimer migration on the (110) surface: (1) concerted jump mechanism and (2) local diffusion anisotropy. During simulations along each path, except path 4, we find that when one adatom is moved (by a series of small displacements) from one stable site to a nearest stable site, another adatom will follow the first one as shown in paths 1, 2 and 3 (see Fig. 1). The two

Table 1 W dimer self-diffusion on the W(ll0), and (211) surfaces Plane

(110) W dimer

W monomer (211) W dimer W monomer

Experiments

Calculations

Ea

Do

Ref.

0.92 + 0.14

1.4 × 10 -3

[4]

0.86 + 0.09

2.1( × 10 ± 2) × 10- 3

[22]

0.82+0.08 "

7.0× 10 -4

[7]

0.83 + 0.02

7.7( x 1.9 ± 1) × 10- 3

[23]

Ed

Do

Path 1 Path 2 Path 3 W monomer

1.38 1.76 1.89 1.14

9.83 1.06 1.10 1.11

10-4 10 -3 10 -3 10 -3

[16]

Dimer in adjacent channels Dimer in one channel W monomer

0.94 1.98 0.79

6.57 × 10- 4 1.84 × 10 -3 1.41 X 10-3

[16]

E d is the activation energy and D O is the pre-exponential factor. (Units: E d in eV and D O in cm2/s.) a Experimental value for W dimer diffusion in adjacent channels.

Ref. × × × X

250

W. Xu, J.B. Adams~Surface Science 339 (1995) 247-257

adatoms will keep a constant bond length ( ~ 2.4 ,~) during the marching process. This d e a r l y indicates W dimers migrate by a concerted j u m p mechanism on the W(110) surface. A s mentioned previously, F I M studies [6,7] found that W dimers are always observed at nearest neighbor positions on the (110) surface, i.e., they do not dissociate into second neighbor dimers; this is consistent with our calculations. Dimer diffusion along path 4 does not occur if we try to move only one atom. In order to approximately determine the energy barrier of this diffusion path, we constrain two adatoms on the bridge sites and let them only relax in the plane perpendicular to the diffusion path. The final result shows that this

e n e r g y ' b a r r i e r is extremely high ( E d = 3.63 eV) compared with other diffusion paths (see Table 1). Table 1 shows the activation energies for three diffusion paths (paths 1, 2 and 3). Compared with paths 2 and 3 with activation energies E d = 1.76 eV and E a = 1.89 eV, respectively, the energy barrier in path 1 is much lower with E a = 1.38 eV. This clearly shows that W dimer diffusion is anisotropic on the (110) surface with the linear displacement (path 1) favored over the re-orientation (path 2 and path 3). Also, the pre-exponential factors D O for three diffusion paths on the (110) surface are calculated using the method described in the Appendix, and are in good agreement with F I M measurements

Table 2 Bond length between adatom and its nearest neighbors in the ground and transition states of W dimer diffusions on the (110), and (211) surfaces (in ,~) (note: we only count the bond with bond length shorter than 4 .~,). Surface

Ground state

Transition state Path 1

Path 2

Path 3

2.49 2.51 2.53 2.53 2.56 2.76 2.84 3.23

2.45 2.51 2.58 2.59 2.60 2.70 2.98

2.46 2.47 2.59 2.59 2.60 2.69 3.02

W(ll0) 2.41 2.55 2.55 2.56 2.56 2.86 2.86 3.29 3.29 Surface

W dimer inside a channel

W dimer in adjacent channels

Ground

Transition

Ground

Transition 1

Transition 2

2.51 2.56 2.56 2.62 2.62 2.63 2.66 2.98 2.98 2.99 2.99 3.33 3.33

2.50 2.50 2.50 2.57 2.58 2.62 2.62 3.17 3.17 3.21 3.26

2.52 2.52 2.54 2.54 2.58 2.58 2.82 2.82 3.21 3.21 3.25 3.25

2.44 2.53 2.54 2.55 2.61 2.64 2.81 2.85 3.00 3.18 3.25 3.36

2.48 2.49 2.51 2.55 2.57 2.58 2.84 2.88 3.17 3.33 3.64 3.67

W(211)

Note: Transition 1 indicates the saddle point with energy barrier Ea ~ 0.23 eV. Transition 2 indicates the saddle point with energy barrier Ed = 0.71 eV.

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W. Xu, J.B. Adams~Surface Science 339 (1995) 247-257

(see Table 1). Experimentally, W dimers are found to move below 300 K by linear displacements, and only at higher temperatures ( ,-, 300 K) do they change their orientation. Thus, our calculations confirm the anisotropy of dimer diffusion observed by FIM. Also, the calculated migration energy for the dimer (1.38 eV) is about 20% higher than the migration energy of a monomer (1.14 eV) [16]. This is qualitatively consistent with FIM observations, which find migration energies of 0.92 and 0.86 eV for the dimer and monomer, respectively. Questions have arisen from experimental and theoretical studies concerning the diffusion mechanism (concerted jump) and diffusion anisotropy. We propose a simple bond coordination model to understand the diffusion mechanism and diffusion anisotropy. In a previous paper, W adatom-adatom interactions on the W(110) surface have been studied [21]. It found there exists a strong attractive interaction ( - 2 . 6 3 eV) for adatom pairs separated at the nearest neighbor sites in the (111) direction. The o bond length between two adatoms is 2.41 A. When adatoms migrate on the (110) surface with an individual jump mechanism, the bond length between adatoms will be stretched to ~ 4 A at the saddle point with one adatom on the bridge site between two substrate atoms. This requires breaking a very strong nearest neighbor bond. The number of first nearest neighbors (NNs) for this adatom pair will change from seven at the ground state to five at the saddle point for the individual jumping mechanism. For the concerted jump mechanism, Table 2 lists the bond lengths between adatoms and their nearest neighbors on the ground state and saddle point for path 1, path 2 and path 3. The coordination number (CN) is nine for an adatom pair in the ground state with seven first NNs and two second NNs. For diffusion along path 1, at the saddle point, there is a large distortion in substrate. All NNs of the adatom pair are displaced (0.25-0.46 ,~) along the migration path of the dimer. The CN changes to eight with seven first NNs and one second NN, i.e., only one second NN bond is broken by this hopping process. Compared with path 1, the CN of path 2 and path 3 decrease from nine to seven at each saddle point with two bonds (one first NN and one second NN) being broken. Therefore, the concerted jump mechanism on the (110) surface is more energetically

<011>

I 1

I I

I I

I

I

I

I

1 0 I

2

3

or-o,, o : o

11 0

-

"-

0 0 l l O l l O I o

0

I I Ii 0

I

0

0

0

0

<111>

0

O

Fig. 2. The binding energies(eV) of W adatom pairs on W(211) surface. Here one adatom of the adatom pair is located at origin. Open circles indicate surface atoms.

favored than the individual adatom jump mechanism. Similarly, the higher CN at the saddle point for path 1 than that for the other two diffusion paths explains the diffusion anisotropy on this surface. 3.2. W dimer on W (211) 3.2.1. Dimer binding sites

The W(211) plane, unlike the (110), is an open surface with channels along the close-packed direction (111) (see Fig. 2). In a previous paper [21], W adatom-adatom interactions on the W ( l l 0 ) surface were studied. Using the same method as in Ref. [21], we have derived the map of adatom-adatom binding energies on the (211) surface for adatom pairs sitting in different sites as shown in Fig. 2. In detail, for adatom pairs inside one channel, i.e, the bond orientation parallel to the close-packed (111) direction, the binding energy is a strong attraction ( - 2 . 3 7 7 eV) for the nearest neighbor site (1, 0), where ( m , n ) is the adatom coordinate with an adatom sitting in the mth binding site along the (111) axis and the nth binding site along the (011) axis from an adatom at (0, 0). The separation between two adatoms is 2.50 ~, which is shorter than the nearest neighbor distance in the bulk structure (,-, 2.73 ,~). When an adatomo moves to a (2, 0) site with a separation of 5.24 A, the interaction changes from attractive to slightly repulsive ( + 0.026 eV). Then the repulsive interactions fall off along this direction. When adatoms are located in adjacent channels, the binding energy changes to repulsion (0.517 eV) at the (0, 1)

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W. Xu, J.B. Adams~Surface Science 339 (1995) 247-257

site. The separation of this adatom pair is 4.42 .~. The interactions between the two adatoms will oscillate as an adatom moves along the adjacent channel. The binding energies change from attractive ( - 0.121 eV) to repulsive (+0.051 eV) at the (1, 1) and (2, 1) sites with separations of 5.32 and 7.07 A, respectively. In general, the anisotropy and oscillatory behavior of W adatom-adatom interactions along a given direction on the W (110) surface [21] are also found on the W(211) surface.

IGLOO

o

3.2.2. Dimer diffusion in one channel To simulate W dimer migration on the (211) surface, two orientations of dimer bonds have been considered in our calculations: (1) an adatom pair closely bound inside a channel with bond length ~ 2.50 .~ and (2) adatoms located in adjacent channels with bond length ~ 5.32 ,~. For W dimer diffusion inside a channel, we find the activation energy is extremely high with E d = 1.98 eV, and the pre-exponential factor D O is equal to 1.84 × 10 -3 cm2/s which is similar to that for monomer diffusion inside one channel (see Table 1). The mechanism for such a migration process is a concerted jump with a constant bond length between the two adatoms ( ~ 2.5 ,~) at all positions. This is in agreement with FIM observations in which closely bound W dimers inside a channel make occasional jumps at very high temperatures (above 420 K) [7], and are always observed together. 3.2.3. Dimer diffusion in adjacent channels When two adatoms sit on adjacent channels, the most stable structure for this adatom pair is shown in Fig. 3a with a bond length of 5.32 A and the orientation of the bond having a 57.3 ° angle to the channel rows. Unlike in-channel dimer diffusion, the diffusion mechanism for adatom pairs in this case is an individual adatom jump. For this adatom pair, there exist four possible diffusion processes as indicated in Figs. 3a to 3d, which will either bring the adatom pair to a shorter bond ~ 4.42 ,~ (a, b) or stretch it to a longer bond ~ 7.07 .~ (c, d): (a) adatom # 1 migrates to its nearest neioghbor site, i.e., the bond length changes to 4.42 A, with an activation energy E d = 0.87 eV; (b) this adatom pair returns to a shorter bond (4.42 ,~) as (a), but with adatom # 2 migrating to its

_•¢'•.8

7

0.64~

1.35"-~ i ~ . ~ 0.64

(a)

(c)

(b)

~ O,17~_~/

0"71

f~1,94

.64

(d)

(e)

o'g14 ~ ~,L~O.17

(3-000 /~0.23 0.64~ (f)

[01 i]

[i-11] ,i

I [111] [2111 Fig. 3. Schematic of W dimer diffusion paths on W(211) surface. The activation energy for migration is also listed.

nearest neighbor site. The energy barrier is very high E d = 1.35 eV; (c) adatom # 1 can stretch this adatom pair to 7.07 .~ by hopping to another nearest neighbor site with E d = 0.94 eV; (d) adatom # 2 can hop to increase the bond length to 7.07 ,~ with an activation energy E d = 0.94 eV. Comparing these activation energies from (a) to (d), we find the most likely jump is (a) with the bond length changed to 4.42 A. Then, after shortening the bond from 5.32 to 4.42 A, there are two diffusion paths for this adatom pair as shown in Figs. 3e and 3f: (e) adatom # 1 (or #2) moves along the [111] direction to its nearest neighbor site with an activation energy E d = 0.71 eV; (f) adatom # 1 (or # 2 ) can migrate in the opposite direction [1]-17. Unexpectedly, the activation energy o

W. Xu, J.B. Adams/Surface

El

=0.87eV

E2 = 0.94 E3=

1.35eV

E4 = 0.94

E5 = 0.71

eV

E7 - E8 = 0.77

E6 = 0.23

eV

E9=El0=0.79eV

(MS11

eV

eV

eV

(MS2)

Fig. 4. Schematic of W adatom pair stable structure (S) and metastable structures (MS1 and MS2). There are ten possible jumps, and the activation energies for those jumps are listed.

is 0.23 eV which is much lower than that of the adatom migration along [ill] in (e). This jump will bring the adatom pair to its original structure as in Fig. 3a. It clearly shows that the diffusion is asymmetric for one adatom of an adatom pair migrating along the channel. We also note that during migration, one adatom will stay in its site while another adatom hops along the channel, i.e., diffusion occurs by an individual jump mechanism. In Fig. 3, the energy barriers for each individual adatom jump have been recorded. However, it does not indicate clearly how a W dimer migrates on the (211) surface. In order to understand how long-range diffusion occurs, we start from the most stable structure (S) of an adatom pair bound in the adjacent channels as shown in Fig. 4. If we consider how an adatom pair moves from a stable structure (S) to another stable structure (or its equivalent one), via the metastable structure (MSl) or (MS2), there are only eight diffusion processes (P,, n = 1, 8) which can be divided into 2 categories: (1) diffusion via the structure MS1 (P,, P2, P3 and P4) and (2) diffusion via the structure MS2 (P,, P6, P,, and Ps). In other words, we ignore those diffusion processes in which

Science 339 (1995) 247-257

253

two adatoms separate over 7.07 A (i.e., dimer dissociates). In the first category, P, includes two individual adatom jumps 1 and 6, and Pz includes jumps 3 and 5 (Fig. 4). These jumps will not result in any net motion for the adatom pair, i.e., adatoms will only oscillate between two structures (S) and (MSl). Long-range diffusion only occurs by P3 and P4, the jump sequences are 1 and 5 and 3 and 6, respectively. Using the method described in the Appendix, we calculate the activation energy Ed and pre-exponential factor D,, for P,, Pz, P3 and P4, Note that P3 and P4 have very similar diffusion rates, but their Ed are much higher than that for a single adatom. Similarly, for jump sequences involving MS2, Ps (jumps 2 and 7) and Ps (jumps 4 and 8) will result in the adatom pair oscillating between two sites, without any net displacement. Long-range diffusion only occurs by P, (jumps 2 and 8) and Pa (jumps 4 and 7). As mentioned above, we will not consider dimer diffusion involving bond dissociation (bond length over 7.07 A). The calculated Ed and D, are listed in Table 3. It shows both short-range (Ps and Ps) and long-range (P, and Pa) jump sequences have the similar diffusion rate. In other words, the probability of an adatom pair staying at one site or migrating away will be roughly the same. Considering all the diffusion processes P,,(n = 1, 8), P1 has the lowest activation energy Ed, i.e., adatom pair vibration will dominate at low temperatures. At moderate temperatures, long-range diffusion will occur by jump processes P7 and Ps, which

Table 3 W adatom pairs self-diffusion in adjacent channels

on the (211) surface with adatoms

Diffusion path

Ed

DO

Pt

0.87 1.83 1.35 1.35 0.94 0.94 0.94 0.94

1.34x 10-s 1.92X1O-3 2.01 x lo- 3 1.41 x10-3 6.57X 10-4 6.42 X lo- 4 6.57X 10-4 6.42X 1O-4

p2 p3

P4 PS ‘6

P, p8

Ed is the activation energy and D, is the pre-exponential factor (units: E,, in eV and D, in cm2/s) for eight diffusion processes P, (n = 1, 8) which are calculated as described in Appendix.

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W. Xu, J.B. Adams~Surface Science 339 (1995) 247-257

are favored over processes P3 and P4. Therefore, the long-range diffusion mechanism is that an adatom pair diffuses via the metastable structure (MS2) by stretching its bond slightly (partial dissociation) and then catching up. However, it is also likely that the dimer will dissociate during this process. The experimental data [2,7,24] indicates that at low temperatures (255 K) the dimer oscillates with the energy barrier ~ 0.76 eV [25], in good agreement with our calculation. However, at moderate temperatures (275 K), the dimer diffuses over long distances without dissociation, and must be heated to higher temperatures (385 K) to dissociate. This is in disagreement with our calculated results, which indicate that the dissociation energy is slightly lower than the migration energy. This discrepancy may be small, but is important. We also carried out calculations which involved the exchange mechanism, in order to consider other possible diffusion paths. One adatom in the dimer separated in adjacent channels will replace one neighboring channel atom and push this atom to become a new adatom. The final configuration of the dimer is that adatoms are still separated in adjacent channels. The calculated energy barriers for such exchange mechanism are always two times larger than that of simple hopping events. Therefore, we conclude the exchange mechanism is unfavored in this case. 3.2.4. Discussion Two major questions are (1) why do dimers in the same channel (the concerted jump mechanism) diffuse slower than those in adjacent channels (the individual adatom jump mechanism) and (2) why do the energy barriers vary significantly for an adatom pair in adjacent channels, depending on the dimer structure. To answer these questions, we examine the bond coordination at the ground state and the saddle point for different structures. For the first question, we find that when two adatoms are closely bound in one channel the coordination number (CN) of the dimer in the ground state is thirteen with seven nearest neighbors (NNs) and six second nearest neighbors (SNNs) (see Table 2). The CN will change to eleven with seven NNs and four SNNs at the saddle point. Two bonds have been broken during the hopping process. On the other hand, for an

adatom pair located in adjacent channels with a bond direction perpendicular to the channel rows, the CN of this adatom pair is twelve (eight NNs and four SNNs) on the ground state. The CN will be the same (twelve) at the saddle points when one adatom hops to its nearest neighbor sites as shown in Fig. 3. As listed in Table 2, the contribution to these energy barriers is due to the stretching of bonds. Therefore, it clearly shows that a concerted jump inside a channel is energetically unfavored over the individual jumps for adatoms in adjacent channels. For the question: why do the energy barriers vary significantly for an adatom pair in adjacent channels, depending on the dimer structure, it is due to (1) very different dimer binding energies in these sites and (2) the difference of bond coordination at the saddle point caused by the asymmetry of the W(211) surface. As described previously, the binding energy for adatom pairs in adjacent channels will oscillate along the (111) direction. From Fig. 2, the most stable binding site for an adatom pair in adjacent channels is the site (1, 1); the neighboring sites (0, 1) and (2, 1) are 0.64 and 0.17 eV higher in energy, respectively. Such variation of the binding energy is closely related to variations in the diffusion barrier when an adatom jumps from one site to another site. Also, the bond coordinations of adatom pairs at the saddle points are different when an adatom jumps along the [111] or [1J-l] directions. For example, as shown in Figs. 3e and 3f, the activation energies are different (0.71 versus 0.23 eV) when adatom # 1 diffuses along the [ i l l ] or the [llT] direction. At the saddle points of two diffusion processes, the CN will be the same as in the ground state (see Table 2). However, compared with adatom diffusion along the [1]'1] direction with only a couple bonds slightly stretched ( < 7%) at the saddle point, two bonds will be greatly enlarged from 3.25 to ~ 3.6 ~, ( ~ 13%) for an adatom hopping along the opposite direction [111]. This bond stretching requires a higher energy.

4. Conclusion

We investigate W dimer self-diffusion on the (110) and (211) surfaces, using a modified fourth moment approximation to tight-binding theory. The

W. Xu, J.B. Adams~Surface Science 339 (1995) 247-257

results described below can be understood in terms of a simple bond coordination model. (1) For a W dimer on the (110) surface, the diffusion mechanism is a concerted jump (both adatoms move simultaneously). Also, linear displacement of the dimer along the (111) direction is favored over bond rotation, in agreement with experiment. The present calculations indicate that the activation energy barrier for W dimer migration is slightly higher than that for single adatom diffusion, which is consistent with FIM observations. However, this model overestimates the energy difference ( ~ 20% higher) compared with FIM studies ( ~ 7% higher). (2) On the (211) surface, when an adatom pair is located inside one channel, the diffusion mechanism is a concerted jump. For an adatom pair in adjacent channels, the dimer migrates through individual adatom jumps. The activation energy is much lower for the adjacent channel case than for one channel case (0.94 versus 1.98 eV), in agreement with the experiment. For the adjacent channel case, the present calculations yield comparable activation energies for dimer diffusion and monomer diffusion, which is in agreement with FIM observations. However, the calculations show that the dissociation energy is slightly lower than the migration energy, which disagrees with experiments. This is a quantitatively small but important flaw of the model.

Acknowledgements This work was supported by the US Department of Energy, Office of Basic Energy Sciences through the Materials Research Laboratory at the University of Illinois under the grant DE-A(O)-76ER01198. We would like to thank the National Center for Supercomputing Application for use of their CRAY-YMP. We are indebted to G. Ehrlich and S.C. Wang for helpful discussions and suggestions.

Appendix. Calculation of the pre-exponential factor D O

W(llO) and W(211) in-channel diffusion. For diffusion on the (110) surface and in-channel diffusion

255

on the (211) surface, the diffusion mechanism is the conceited jump with two adatoms jumping simultaneously. As described previously, the bond length between two adatoms keeps nearly unchanged ( ~ 2.5 ,~) during these jumps with a strong attractive interaction between two adatoms ( ~ 2 eV). Therefore, we can simplify the calculation by ignoring the inter-vibration of two adatoms. Similar to a single adatom diffusion on the surface [16], the D O of an adatom pair diffusion can be expressed by nul 2

Do=

Z---d-

(A.1)

Here n is the number of jump directions available to the adatom pair, u is the attempt frequency which can be calculated using a simple harmonic oscillator model, l is the jump length to the adjacent site and d is the dimensionality of the surface (1 for channelled and 2 for normal diffusion). For the (211) surface, the n is equal to 2 for adatom pair in-channel diffusion. For the (110) surface, the adatom pair diffusion primarily along the (111) direction with linear displacement (n = 2 and d = 1). Occasionally, it will rotate (n = 4 and d -- 2). In this simple model, the migration entropy is assumed to be zero; this estimate of D O is probably correct to within a factor of 2 [9]. W(211) adjacent channels diffusion. In contrast to an adatom pair in-channel diffusion on the (211) surface, the diffusion mechanism for an adatom pair in adjacent channels on the (211) surface is an individual jump of adatoms along the channel rows. Such simplification as described above, therefore, cannot be used in this case. The diffusion coefficient D, in general, can be written as

12F O = -~,

(A.2)

where l is the jump length of an adatom pair from a stable site to adjacent stable site, d is the dimensionality (1 for this case), F is the jump rate of an adatom pair from one stable structure to another equivalent stable structure. As described in text, the most stable structure (S) of an adatom pair in adjacent channels has a bond length 5.32 .A as shown in Fig. 4. When an adatom pair diffuses from one stable structure to another

256

W. Xu, J.B. Adams~Surface Science 339 (1995) 247-257

equivalent stable structure, it has to change to one of the metastable structures (MS1) or (MS2) with a bond length of 4.42 or 7.07 ,~, respectively (see Fig. 4). To calculate the jump rate F of an adatom pair, one must determine the single adatom jump rate k along the channel. Considering the energy barriers of different jumping processes, as shown in Fig. 4, there are four different jump rates when an adatom pair stays in the most stable structure (S), i.e., ka, k2, k3, and k 4 for two adatoms, respectively. Similarly, there are two jump rates, k 5 and k6, for one metastable structure (MS1) and also four jump rates, kT, ks, k9, and kl0 (here k 7 = ks, k 9 = kl0) , for another metastable structure (MS2). Any jump rate, k, can be expressed as

where k, (n = 1, 10) is the rate constant as described as above. As an example, we will show how to calculate the jump rate F 1 for the diffusion event P1. The probability of an adatom jumping from the stable structure (S) to the metastable structure (MS1) via path 1 is

k = v exp - - - ~

F1.

(om)

,

(A.3)

where v is the vibrational frequency of an adatom at a stable site, and G m is the free-energy change when an adatom hops from a stable site to a saddle point position. At zero pressure with the assumption of zero activation entropy, the free-energy G m is equivalent to the activation energy E d. There are totally 10 jump rates k,, (n = 1, 10) for a single adatom which will change the structure of an adatom pair as shown in Fig. 4. As mentioned in the text, there are eight diffusion processes Pn (n = 1, 8) for an adatom pair if we only consider its jumps from a stable structure (S) to another stable structure (or its equivalent one) via the metastable structure (MS1) or (MS2). Hence, the number of the adatom pair jump rate F will be 8. To calculate the jump rate F , we have to know how long it will take for an adatom jump to occur. The average time for one jump event to occur at the stable structure (S), the metastable structure (MS1) and the metastable structure (MS2) are "./'1, T2 and "./-3 which can be written as 1 (A.4)

3-1 = ka + k2 + k3 + k4 , 1 T2

"./-3 "~--

(A.5)

2 ( k 5 + k 6) ' 1

1

k 7 + k 8 + k 9 -+- kl0

2 ( k 7 + k9) '

(A.6)

(A.7)

X1 = klTl"

Similarly, the probability of an adatom hopping from the metastable structure (MS1) to the original structure (via path 6) is

(A.8)

X2 = k63-2.

The total jump rate is X1,1(2

.

.

"./'1 + "./-2

.

k13-1k63-2

(A.9)

T1 "j- T2

Using the same analysis, F for other diffusion processes can be calculated as follows: k33-1k53- 2

F2 = - - ,

3-1 "+ T2

k13-1 k5 "./-2

F3= - - ,

(A.10)

(A.11)

"./'1 + T2 k33-1k63- 2

F4 = - - ,

"./'1 At- T 2

k2T1k773

F5 = - - ,

(A.12)

(A.13)

T1 -+- './-3 k43-1k83- 3

F6 = - - ,

(A.14)

T 1 -~- T 3

r 7

k23-1k8T 3

- - ,

(A.15)

T1 + "/'3

F8

k43-1 k73- 3 ,7"1 q- T 3

(g.16)

By substituting F n (n -- 1, 8) into Eq. (A.2), we can derive the diffusion coefficient /9, (n = 1, 8) as a function of temperature (T). By plotting the In D, versus 1/T curve for each diffusion process, the activation energy E d and the pre-exponential factor D O can be derived as listed in Table 3.

W. Xu, J.B. Adams/Surface Science 339 (1995) 247-257

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[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

257

P. Blandin and C. Massobrio, Phys. Rev. B 47 (1993) 13687. K.D. Shiang and T.T. Tsong, Phys. Rev. B 49 (1994) 7670. P.J. Feibelman, Phys. Rev. Lett. 58 (1987) 2766. G. Ehrlich, Surf. Sci. 246 (1991) 1, and references therein. Wei Xu and J.B. Adams, Surf. Sci. 301 (1994) 371; 311 (1994) 441. Wei Xu and J.B. Adams, Surf. Sci. 319 (1994) 45. Wei Xu and J.B. Adams, Surf. Sci. 319 (1994) 58. P.G. Flahive and W.R. Graham, Surf. Sci. 91 (1980) 449. N.A.W. Holzwarth, J.A. Chervenak, C.J. Kimmer, Y. Zeng, W. Xu and J.B. Adams, Phys. Rev. B 48 (1993) 12136. G. Vineyard, J. Phys. Chem. Solids 3 (1957) 121. A.F. Voter and J.D. Doll, J. Chem. Phys. 80 (1984) 5832. W. Xu and J.B. Adams, Surf. Sci. 339 (1995) 241. D.W. Bassett and M.J. Parsley, J. Phys. D 3 (1970) 707. S.C. Wang and G. Ehrlich, Surf. Sci. 206 (1988) 451. G. Ehrlich, private communication. Estimated from the onset temperature in FIM experiments.